Started a new job recently, I have kept my old job and they are now asking for a reference from my new place of work. Am I required to give it to them?

Hi all, as the title says, I have started a new job ~1mo ago, and as I was working at my previous place on a bank contract, I have decided to stay on there as bank, so no change in contract, but I am working less hours than previous. I obviously had to give references when applying to my new job, all fine and no issues.

My old job have just emailed me to ask for references from my new job for 'HR checks'. I required no references when starting my old job, so I don't know why they would ask for some now, especially from an employment which was started after when I started there.

I work in healthcarein case that changes anything. Thanks